Consider the problem given in Example 14. 7 for the case when the reaction is \(2 \mathrm{~A} ightarrow \mathrm{B}\) and the reaction is second order with respect to A. Derive an expression, similar to Equations (14.19) and (14.20), for the optimal cycle time in terms of the relevant parameters.

Example 14.7

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Determine the optimal batch reaction and cycle time for the first-order, irreversible reaction A-B using the following data and plot the OF [Equation (14.17)] as a function of 0. V = 5 m Cao=2 lmol/m k = 0.3173 h- Substituting into Equation (14.18) yields y = $400/kmol Cclean = $1200 41=xVC40-Cclean (700) (5) (2)-1200 = 5800 B = (x+y) VC0 = (400+700) (5) (2) = 11,000 C = 1clean = 2 clean = 2h x = $700/kmol - (BkC Bikt opt + B), ktopt A = [(11.000) (0.3173)(2)+(11.000) (0.3173) opr +11.000]e-0.3173topt - 5800 = 0 (6980 +3490topr+11,000)e-0.3173fopr-5800 = 0